Why Read This Book? - Index of

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(a) limx→a f(x) =−∞ (b) limx→a + f(x) =+∞ (c) limx→a− f(x) =+∞ (d) limx→+∞ f(x) =+∞ (e) limx→+∞ f(x) =−∞ (f) limx→−∞ f(x) =+∞ (g) limx→−∞ f(x) =−∞ 7.4 Limits Involving Infinity 223 EXERCISE 7.4.10 Given a function f , limx→a f(x) =+∞if and only if lim f(x) = lim f(x) =+∞ (7.15) x→a + x→a− What other kinds of theorems can we expect for limits of infinity? Can we prove something that resembles Theorem 7.4.4? The answer is yes, but the theorems will look different because the limits are not necessarily real numbers. Consequently, we have to begin with Definition 7.4.7 and do some of the work from scratch. The parts of the next exercise are ordered so that some of the later ones follow quickly from the earlier ones. This should make your work a little more efficient. These results and those in Exercise 7.4.12 could just as easily be stated and proved in terms of one-sided limits. EXERCISE 7.4.11 Suppose f(x) → L, g(x) →+∞, and h(x) →+∞as x → a. Then as x → a, (a) −g(x) →−∞ (b) 1/g(x) → 0 (c) f(x) + g(x) →+∞ (d) f(x)g(x) →+∞if L>0 (e) f(x)g(x) →−∞if L
224 Chapter 7 Functions of a Real Variable EXERCISE 7.4.12 Suppose there exists a deleted neighborhood of a such that f(x) > 0 for all x in the deleted neighborhood. Suppose also that f(x) → 0 as x → a. Then 1/f(x) →+∞. Similarly, if f(x) < 0 for all x in the deleted neighborhood and f(x) → 0, then 1/f(x) →−∞. In the same way that we use the notation x → a + and x → a − to mean x approaches a from the right and left, respectively, we can create a shorthand notation for functions that behave as in Exercise 7.4.12. We write f(x) → 0 + to mean that f approaches zero and is positive on a deleted neighborhood of a, and we say f approaches zero through positive values. Similarly we write f(x) → 0 − to mean that f approaches zero and is negative on a deleted neighborhood of a, and we say f approaches zero through negative values. EXERCISE 7.4.13 Let f(x) = (x 2 − 4)/(x − 1) for all real numbers x �= 1. Find with verification lim x→1 − f(x) and lim x→1 + f(x). 7.5 Continuity The word continuous may already be a part of your mathematical vocabulary. Perhaps your calculus class delved into continuity enough to provide an ɛ-δ definition. More than likely, your notions of continuity are probably best summarized as a belief that a continuous function can be sketched without lifting your pencil off the paper. The graph is one clean, easily drawable piece. Even though such a view can be helpful in your understanding of some characteristics of continuity, it is far from true that all continuous functions are so easily drawable. The bizarre examples of undrawable continuous functions will come later in your study of mathematics. For now, we define the terms and study the basic results. We begin with continuity at a single point in the domain, and then we talk about continuity on a subset of the domain. In the same way that we talk about left-hand and right-hand limits, we will then talk about left continuity and right continuity. 7.5.1 Continuity at a Point If limx→a f(x) exists, it describes the behavior of f near a but not at a. It is possible that limx→a f(x) = L, while f(a) might either fail to exist or exist and be different from L. Ifa is in the interior of the domain of f , if limx→a f(x) exists, and if this limit is the value of f(a), we give this phenomenon a name. Definition 7.5.1 Suppose f : S → R is a function and a ∈ Int(S). We say that f is continuous at a provided lim f(x) = f(a) (7.16) x→a
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
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Solution 1.4 Negations of Statement
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Solution 1. Everyone in this class
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1.5 How We Write Proofs 41 Our purp
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1.5 How We Write Proofs 43 the nega
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Properties of Real Numbers 2 It’s
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2.1 Basic Algebraic Properties of R
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2.1.2 Properties of Multiplication
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2.2 Ordering Properties of the Real
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(c) For all real numbers a, a2 ≥
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2.3 Absolute Value 55 (⇐) Now sup
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2.4 The Division Algorithm 57 mn =
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2.5 Divisibility and Prime Numbers
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2.5 Divisibility and Prime Numbers
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Sets and Their Properties 3 All of
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U A B Figure 3.5 Venn diagram illus
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(f) For all sets A, B, and C,ifA
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3.2 Proving Basic Set Properties 69
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3.3 Families of Sets 71 Notice that
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Example 3.3.2 Consider the family o
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3.3 Families of Sets 75 x ∈ � A
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3.3 Families of Sets 77 and ... and
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Example 3.4.2 For any positive inte
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3.4 The Principle of Mathematical I
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Proof. To clean up the notation, we
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3.5 Variations of the PMI 85 EXERCI
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(g) (a −m ) −n = a mn (h) (ab)
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Strong Induction 3.5 Variations of
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3.6 Equivalence Relations 91 EXERCI
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3.6 Equivalence Relations 93 beginn
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3.6 Equivalence Relations 95 constr
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3.7 Equivalence Classes and Partiti
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3.8 Building the Rational Numbers 1
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3.8 Building the Rational Numbers 1
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3.10 Irrational Numbers 107 EXERCIS
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3.10 Irrational Numbers 109 To the
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EXERCISE 3.10.2 √ 2 is irrational
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(a) {(x, y) : x ≤ y} (b) {(x, y)
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3.11 Relations in General 115 (O3)
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3.11 Relations in General 117 at al
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Functions 4 Second only to sets, fu
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4.1 Definition and Examples 121 pro
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Solution We show that T satisfies p
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4.2 One-to-one and Onto Functions 4
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4.2 One-to-one and Onto Functions 1
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A 1 x A B f (A 1 ) Figure 4.4 The i
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4.4 Composition and Inverse Functio
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4.4 Composition and Inverse Functio
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4.5 Three Helpful Theorems 135 The
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4.6 Finite Sets 137 EXERCISE 4.5.3
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4.7 Infinite Sets 139 The next exer
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4.7 Infinite Sets 141 of A. This or
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4.7 Infinite Sets 143 EXERCISE 4.7.
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EXERCISE 4.8.3 If A1,A2, and B are
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4.8 Cartesian Products and Cardinal
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4.8 Cartesian Products and Cardinal
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4.9 Combinations and Partitions 151
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4.10 The Binomial Theorem 157 EXERC
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EXERCISE 4.10.3 Determine the follo
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(c) The coefficient of ab 3 c 2 in
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P A R T I I Basic Principles of Ana
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The Real Numbers 5 Let’s look at
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5.1 The Least Upper Bound Axiom 167
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5.2 The Archimedean Property 169 EX
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5.2 The Archimedean Property 171 No
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Page 202 and 203: 5.5 Closure of Sets 179 The constru
Page 204 and 205: 5.6 Compactness 181 EXERCISE 5.6.3
Page 206 and 207: 5.6 Compactness 183 EXERCISE 5.6.13
Page 208 and 209: Sequences of Real Numbers 6.1 Seque
Page 210 and 211: 6.1 Sequences Defined 187 Example 6
Page 212 and 213: EXERCISE 6.1.15 Consider the sequen
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Page 216 and 217: 6.2 Convergence of Sequences 193 EX
Page 218 and 219: 6.2 Convergence of Sequences 195 To
Page 220 and 221: 6.3 The Nested Interval Property 19
Page 226 and 227: a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
Page 228 and 229: Example 6.4.7 The terms a0 = 1 a1 =
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Page 232 and 233: 7.1 Bounded and Monotone Functions
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Page 240 and 241: 7.3 More on Limits 217 values of 0
Page 242 and 243: 7.4 Limits Involving Infinity 219 W
Page 244 and 245: 7.4 Limits Involving Infinity 221 T
Page 248 and 249: 7.5 Continuity 225 If f is not cont
Page 250 and 251: 1 2 3 4 5 6 Figure 7.6 Some example
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Page 254 and 255: 7.6 Implications of Continuity 231
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Page 258 and 259: 7.7 Uniform Continuity 235 we decla
Page 260 and 261: 7.7 Uniform Continuity 237 Next, le
Page 262 and 263: 7.7.2 Uniform Continuity and Compac
Page 264 and 265: P A R T I I I Basic Principles of A
Page 266 and 267: Groups 8 In its simplest terms, alg
Page 268 and 269: 8.1 Introduction to Groups 245 are
Page 270 and 271: 8.1 Introduction to Groups 247 Defi
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Page 274 and 275: Show that C × is an abelian group
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Page 278 and 279: 8.2 Subgroups 255 Notice how proper
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Page 282 and 283: EXERCISE 8.2.23 If G is a group and
Page 284 and 285: 8.3 Quotient Groups 261 [0] ={...,
Page 286 and 287: 8.3 Quotient Groups 263 is standard
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2 3 1 3 1 Figure 8.1 Rigid square u
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EXERCISE 8.4.12 Complete Table 8.64
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8.5 Normal Subgroups 277 Theorem 8.
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8.5 Normal Subgroups 279 point to b
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8.6 Group Morphisms 281 EXERCISE 8.
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8.6 Group Morphisms 283 Notice that
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Ker � e x 2 x maps to x Ker � G
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Rings 9 We can create algebraic str
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9.1 Rings and Fields 289 (R9) Multi
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9.1 Rings and Fields 291 we define
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9.2 Subrings 293 In addition to pro
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9.2 Subrings 295 p1 = q1 = 3. Verif
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9.3 Ring Properties 297 Theorem 9.3
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9.3 Ring Properties 299 EXERCISE 9.
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9.4 Ring Extensions 301 in the inte
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9.4 Ring Extensions 303 Example 9.4
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9.4 Ring Extensions 305 We insist t
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9.5 Ideals 307 An ideal, say a left
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9.6 Generated Ideals 309 Proof. Sup
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EXERCISE 9.6.6 In the integers, wha
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9.7 Prime and Maximal Ideals 313 Ev
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9.8 Integral Domains 315 EXERCISE 9
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9.8 Integral Domains 317 Corollary
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9.9 Unique Factorization Domains 31
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9.10 Principal Ideal Domains 321 al
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9.10 Principal Ideal Domains 323 So
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9.11 Euclidean Domains 325 next exe
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9.11 Euclidean Domains 327 Exercise
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Proof. Let f and g be nonzero polyn
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9.12 Polynomials over a Field 331 W
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9.13 Polynomials over the Integers
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9.14 Ring Morphisms 335 Example 9.1
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9.14 Ring Morphisms 337 EXERCISE 9.
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9.15 Quotient Rings 339 this, howev
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9.15 Quotient Rings 341 Because the
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9.15 Quotient Rings 343 However, th
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Index =,51 ɛ (epsilon), 167-168 ɛ
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Index 347 Cosets, 263, 264, 265 Lag
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Index 349 Gauchy sequences, 202-206
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Index 351 identity, 124 relations v
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Index 353 Quaternions, 248, 253, 25
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Index 355 Tower of Hanoi, 84 Transc
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